Yeah this has been addressed before I believe. I think the server-side latency is 0.40 seconds. That's pretty massive. Compared to server-side latency of (The one the video I saw used) Heroes of Newerth, which was 0.20 seconds from, I think East Coast to Europe, and 0.02 seconds from East Coast to West Coast servers (Ergo Heroes of the Storm based on that guy's tests has like 200x more server-side latency).
0.40 is not 200 times 0.02.
Also, it's impossible for East Coast - West Coast latency to be 0.02 seconds. At the speed of light, it would take about 0.013 seconds to get from New York to Los Angeles along the minimal-distance geodesic. With router latency and the fact that signals don't actually propagate at the speed of light through networks, you're looking at at least around 0.05 seconds latency in practice.
For real data: AT&T reports average network latencies between their network hubs of 0.065 seconds (or so) between LA and NYC. Source: http://ipnetwork.bgtmo.ip.att.net/pws/global_network_avgs.html
Consumers will have higher latency than this because they are not connected directly at a major internet routing station (and usually the server they're connecting to isn't, either) - usually between 0.07 and 0.08 seconds, at least.
Beyond that, human reaction time to simple stimulus that you expect is on average about 0.2 seconds (where "expect" means something like: you are given a button, and a light. You are told to press the button when the light turns on, which will happen at a random time). To less-expected stimuli with more complicated possible reactions, it can be significantly slower.
So latency of 0.40 seconds isn't usually
quite as bad as it sounds (it's more like tripling your normal reaction time, not multiplying it by 20), but it's still very noticeable (0.02 latency, on the other hand, had might as well be 0 latency for a human being).
very significant in situations where you can anticipate
when you need to press a button - i.e. pressing an escape when you leave a stun, as you mentioned. A human might often have close to 0 delay in this type of situation, so 0.4 seconds of latency becomes a much bigger factor.
Heroes definitely feels notably less responsive than LoL for example though.